We shall denote the collection of characteristic functions of elements of \(\operatorname{\mathcal {O}_{ad}}\)  as \(\mathcal {U}_{ad}\) , i.e., \(\mathcal {U}_{ad}=\lbrace \chi _\Omega ; \Omega \in \operatorname{\mathcal {O}_{ad}}\rbrace \) , and whenever we mention \(\mathcal {U}_{ad}\)  we take into account the weak\(^*\)  topology in \(L^\infty (D)\) .
We refer the reader to [1]} and [2]} for a more detailed discussion on the topology of characteristic functions of finite measurable domains.
